PROOF 80
Prove.
The additive inverse is unique
On the integers ℤ.
A way to read this proof
First separate what is assumed from what is to be proved. At each step, ask which definition or earlier result permits the move. If it is unclear, return to that premise rather than skipping to the conclusion.
Definition D1
That x is an additive inverse of a means x + a = 0.
Proof.
Here 0 is the identity of A1. Whether an inverse exists is not asked.A1
∎
Theorem T
If x and y are both additive inverses of a, then x = y
Proof.
(1)x + a = 0 and y + a = 0assumption, D1
(2)x = x + 0A1
(3) = x + (y + a)(1)
(4) = x + (a + y)A3
(5) = (x + a) + yA2
(6) = 0 + y(1)
(7) = y + 0A3
(8) = yA1
∎
Corollary C
Writing the additive inverse of a as −a is justified.
Proof.
(1)If a has an additive inverse, it has only one.T
(2)If there were two or more, which of them −a refers to would be undetermined.D1, convention
(3)Therefore −a refers to a single number.(1), (2)
∎
Remark
Can A2 be removed from this proof?
The two inverses x and y first meet at (5). Check what that line leans on.